Your criteria matches just about all of Fort Mill and Tega Cay - most (if not all) subdivisions are family friendly, close to shopping, with backyards and HOAs. You will likely only find a 3 bedroom for $1800 though, or a very dated 4-bedroom.
Muse Realty handles a lot of rentals in our area - here is a page of current listings:
AVAILABLE RENTALS
Most listings are in Rock Hill, but they do have a few in Fort Mill and Tega Cay.
As for Clover and Indian Land, in Clover it's a mix of country and built-up subdivisions. Indian Land is just about as built up as Fort Mill.
My suggestion is contact a realtor (or Muse) that deals with rentals and tell them what you want. In any of those areas, you can't go wrong.